Plotting the Quadratic Function Using Parameters a, b and c

🏆Practice parameter functions and graph plotting

Plotting the graph of the quadratic function and examining the roles of the parameters \(a, b, c\) in the function of the form \(y = ax^2 + bx + c\)

The quadratic function has three relevant characteristics:

aa – the coefficient of X2X^2.
bb – the coefficient of XX.
cc – the constant term.

Steps to graph a quadratic function –

  1. Let's examine the parameter aa and ask: Is the function upward or downward facing?
  2. Let's find the vertex of the function using the formula and then find the Y-coordinate of the vertex.
  3. Let's find the points of intersection with the XX axis by substituting (Y=0Y=0).
  4. Let's draw a coordinate system and first mark the vertex of the parabola.
    Then, let's examine if the function is smiling or crying and mark the points of intersection with the XX axis that we found. Draw accordingly.
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Test yourself on parameter functions and graph plotting!

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\( y=-2x^2+3x+10 \)

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Graphing the quadratic function ausing parameters a, b and c

Let's present the quadratic function and understand the meaning of each parameter in it.
y=ax2+bx+cy=ax^2+bx+c

aa – the coefficient of X2X^2
    - -determines if the parabola will be a maximum or minimum parabola (sad or smiling)   - determines the steepness – its width.

aa must be different from 0.
If aa is positive – the parabola is a minimum parabola – smiling
If aa is negative – the parabola is a maximum parabola – sad
The larger aa is – the narrower the function will be and vice versa.

bb - the coefficient of XX
     can be any number
indicates the position of the parabola along with CC.
determines the slope of the function at the intersection point with the YY axis
(not relevant to the material) 

cc – the constant term
    is not dependent on XX
can be any number
indicates the position of the parabola along with XX
determines the y-intercept
responsible for the vertical shift of the function


Wonderful. Now, let's move on to plotting the quadratic function.

Steps to graph a quadratic function –

  1. Let's examine the parameter aa and ask: is the function upward or downward facing?
  2. Let's find the vertex of the function using the formula and then find the Y-coordinate of the vertex.
  3. Let's find the points of intersection with the XX axis by substituting (Y=0Y=0).
  4. Let's draw a coordinate system and first mark the vertex of the parabola.
    Then, let's examine if the function is smiling or sad and mark the points of intersection with the XX axis that we found. Draw accordingly.

Let's see an example -

In the following function:
Y=−4X2+4x+3Y=-4X^2+4x+3

  1. a=−4a=-4, negative. Therefore, the function is downward facing.
  2. Let's find the vertex of the parabola:
    X=−4−4∗2=−4−8=12X=\frac{-4}{-4*2}=\frac{-4}{-8}=\frac{1}{2}
    y=−4∗122+4∗12+3y=-4*\frac{1^2}{2}+4*\frac{1}{2}+3
    y=4y=4
    Vertex of the parabola (12,4) (\frac{1}{2}, 4)  
     
  3. Let's find the intersection points with the XX axis
    Substitute
    Y=0Y=0
    and we get:
    −4X2+4x+3=0-4X^2+4x+3=0
    X=−12,112 X=-\frac{1}{2},1\frac{1}{2} 
     
  4. Draw a coordinate system, mark relevant points, and draw logically:
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